What are the red (T) and numbers?

These are 'Traits' and bonuses.
The red (T) indicates the trait your horse has.
The bonus numbers are also shown in red like this for example: +13 on skill and 10% on specialization.
The specs trait means that your horse has a 10% bonus for a certain show type.
The skill trait indicates the bonus your horse has on a certain skill.
These bonuses are added to whatever skills/specs your horse has.
On store horses you can set these.
When a new foal is born, it will inherit the way the skills are shared on its parents or the best fit.

The specialization trait of a foal is determined by a 33% chance that it will inherit the 'best spec' trait, 33% chance it will inherit the mare's spec. trait and 33% chance it will inherit the stallion's spec. trait. The 'best spec' would be determined by adding the specialization %'s of the parents and it being the highest % sum.